Stanley S. Wainer

ORDINAL RECURSION, AND A REFINEMENT OF THE EXTENDED GRZEGORCZYK HIERARCHY

It is well known that iteration of any number-theoretic function f , which grows at least exponentially, produces a new function f ′ such that f is elementary-recursive in f ′ (in the Csillag-Kalmar sense), but not conversely (since f ′ majorizes every function elementary-recursive in f ). This device was first used by Grzegorczyk [3] in the construction of a properly expanding hierarchy {ℰ n : n = 0, 1, 2, …} which provided a classification of the primitive recursive functions. More recently it was shown in [7] how, by iterating at successor stages and diagonalizing over fundamental sequences at limit stages, the Grzegorczyk hierarchy can be extended through Cantors second number-class. A problem which immediately arises is that of classifying all recursive functions, and an answer to this problem is to be found in the general results of Feferman [1]. These results show that although hierarchies of various types (including the above extensions of Grzegorczyks hierarchy) can be produced, which range over initial segments of the constructive ordinals and which do provide complete classifications of the recursive functions, these cannot be regarded as classifications “from below”, since the method of assigning fundamental sequences at limit stages must be highly noneffective. We therefore adopt the more modest aim here (as in [7], [12], [14]) of characterising certain well-known (effectively generated) subclasses of the recursive functions, by means of hierarchies generated in a natural manner, “from below”.

Slow Growing Versus Fast Growing

I falsely claimed, as an aside remark in [8] and also implicitly in the abstract [9], that the slow-growing hierarchy “catches up” with the fast-growing hierarchy at level Γ 0 , i.e. that, for all x > 0, where x ′ is some simple (even linear) function of x . Girard [4] gave the first correct analysis of the deep relationship which exists between G and F , based on his extensive category-theoretic framework for -logic. This analysis indicates that the first point at which G catches up with F is the ordinal of the theory ID ω (0 of arbitrary finite iterations of an inductive definition. This is very far beyond Γ 0 ! In particular, in order to capture F at level ∣ID n ∣ the slow-growing hierarchy must be generated up to ∣ID n +1 ∣, i.e. one extra iteration of an inductive definition is needed in order to generate sufficient new ordinals.

Ordinal Bounds for Programs

In this paper we use methods of proof theory to assign ordinals to classes of (terminating) programs, the idea being that the ordinal assignment provides a uniform way of measuring computational complexity “in the large”. We are not concerned with placing prior (e.g. polynomial) bounds on computation-length, but rather with general methods of assessing the complexity of natural classes of programs according to the ways in which they are constructed. We begin with an overview of the method in section 2, the crucial idea being supplied by Buchholz’s ω+–rule. Section 3 introduces a large class of higher-order programs based on Plotkin’s PCF, but with “bounded” fixed point operators controlled by given well-ordering. A Tait-style computability proof then ensures termination. In section 4 the details of the ordinal assignment method are worked out for the case where the given well-ordering is just the ordering of the natural numbers. The complexity bounds thus obtained turn out to be the slow-growing functions G α with α below the Bachmann-Howard ordinal. Thus the functions computed by PCF<–programs are just the provably recursive functions of arithmetic.

Computability - Logical and Recursive Complexity

The basis of this short course is the strong analogy between programs and proofs (of their specifications). The main theme is the classification of computable number-theoretic functions according to the logical complexity of their formal specification or termination proofs. A significant sub-branch of mathematical logic has grown around this theme since the 1950’s and new ideas are presently giving rise to further developments. The methods employed are chiefly those from proof theory, particularly “normalization” as presented in the accompanying lectures of Helmut Schwichtenberg, and “ordinal assignments”. Since program-termination corresponds to well-foundedness of computation trees, it is hardly surprising that transfinite ordinals and their constructive representations play a crucial role, measuring the logical complexity of programs and of the functions which they compute.

Equational derivation vs. computation

Abstract Subrecursive hierarchy classifications are used to compare the complexities of recursive functions according to (i) their derivations in a version of Kleenes equation calculus, and (ii) their computations by term-rewriting. In each case ordinal bounds are assigned, and it turns out that the respective complexity measures are given by (i) a version of the Fast Growing Hierarchy, and (ii) the Slow Growing Hierarchy. Known comparisons between the two hierarchies then provide ordinal trade-offs between (i) derivation and (ii) computation. Characteristics of some well-known subrecursive classes are also read off.

Proof theory: a selection of papers from the Leeds Proof Theory Programme 1990

Preface Programme of lectures 1. Basic proof theory S. Wainer and L. Wallen 2. A short course in ordinal analysis W. Pohlers 3. Proofs as programs H. Schwichtenberg 4. A simplified version of local predicativity W. Buchholz 5. A note on bootstrapping intuitionistic bounded arithmetic S. Buss 6. Termination orderings and complexity characterisations E. Cichon 7. Logics for termination and correctness of functional programs, II. Logics of strength PRA S. Feferman 8. Reflecting the semantics of reflected proof D. Howe 9. Fragments of Kripke-Platek set theory with infinity M. Rathjen 10. Provable computable selection functions on abstract structures J. Tucker and J. Zucker.

Program Transformation and Proof Transformation

A ”linear — style” sequent calculus makes it possible to explore the close structural relationships between primitive recursive programs and their inductive termination proofs, and between program transformations and their corresponding proof transformations. In this context the recursive — to — tail — recursive transformation corresponds proof theoretically to a certain kind of cut elimination, called here ”call by value cut elimination”.

THE 1-SECTION OF A COUNTABLE FUNCTIONAL

The continuous or countable functionals were independently denned by Kleene [9] and Kreisel [10]. They were intended as a suitable basis for constructive mathematics, and thus it is interesting to investigate various notions of recursion on the countable functionals. There have been two main streams in this investigation, the study of countable recursion and the study of computability or Kleene-recursion. Countable recursion is the theory of recursion on the associates. Gandy and Hyland [3] and Hyland [7] are good sources for the recent development of countable recursion. This paper will mostly be concerned with Kleene-recursion on the countable functionals as denned in Kleene [8] and [9]. We assume some familiarity with the countable functionals and associates, as presented in Kleene [9], Bergstra [1] or any other paper on the subject. Pioneering work with recursion in nonnormal objects was done by Grilliot [4], who proved that a functional F of type 2 is normal if and only if its 1-section (that is the set of functions recursive in F ) is closed under ordinary jump, and if and only if F is continuous on 1-section ( F ). Hinman [6] constructed a countable functional that is not recursively equivalent to a function, and thereby showed that recursion in nonnormal functionals is an extension of ordinary recursion in functions. In [6], Hinman asked if there are functionals with topless 1-sections, i.e. with no maximal elements in the semi-lattice of degrees. This was answered in the affirmative by Bergstra [1], using a spoiling construction. Thus the class of 1-sections of functionals extends the class of 1-sections of functions.

The Proof Theoretic Complexity of Recursive Programs

We explore, from the point of view of mathematical logic, the structural relationships between recursive programs and their termination proofs, and between program transformation and proof transformation, particularly in the case of the recursive- to-tail-recursive transformation. Ideas from proof theory and recursion theory are applied, so as to measure in an abstract way, the logical cost of such transformations.

Four Lectures on Primitive Recursion

The history of primitive recursion traces back to Dedekind, Hubert, Godel, Ackermann, Herbrand, Peter and Kleene. Some may consider it ‘old-hat’ but it is no less important for all that. The aim in these lectures is merely to present the basic ideas and four different characterizations, illustrating fundamental connections with complexity, term-rewriting and proof theory.